Explicit multi-step peer methods for special second-order differential equations

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Abstract

The construction of s-stage explicit two- and three-step peer methods of order p=2s and p=3s is considered for the solution of non-stiff second-order initial value problems where the right-hand side does not depend on y′. By numerical search with respect to large stability intervals and small error constants sequential and parallel methods are constructed. Numerical tests of these peer methods in MATLAB and comparisons with a Runge–Kutta–Nyström method show the efficiency of the proposed methods.

Introduction

In this paper we consider explicit two- and three-step peer methods for the solution of second-order differential equations where the right-hand side does not depend on y′. Second-order differential equations appear in many applications, especially in physical problems. For problems with negligible friction one often has to solve the special class of equations considered below. Astronomical problems are particularly interesting because the solutions of these problems are smooth and high accuracy is required, e.g. see [1]. The special form of the considered class of second-order differential equations makes it possible to obtain methods of higher order than in the case of general first-order systems (for the same number of stages). Popular methods are the Runge–Kutta–Nyström methods which are generated from Runge–Kutta methods after application of Runge–Kutta methods to a rewritten first-order system and utilization of the absence of a y′-term on the right-hand side. See [2] for the description of a robust, reliable and efficient code implementing Runge–Kutta–Nyström methods. We use a sixth-order Runge–Kutta–Nyström method [3] for comparison with the peer methods.

Explicit peer methods for the solution of first-order differential equations demonstrated their efficiency in [4], [5], [6], [7]. The peer property means that all stages of the method have similar properties, for instance, the same order. Hence, these methods may easily provide dense output. A subclass of these methods also allows parallel implementation. In this paper we will devise peer methods for a class of second-order equation.

This paper is organized as follows: In Section 2 we formulate the classes of explicit two- and three-step peer methods.

In Section 3 we consider the theory of multi-step peer methods, addressing order conditions, stability and convergence. By a special ansatz we construct methods which have optimal zero stability.

Implementation issues are discussed in Section 4, for instance, how to avoid singular matrices which could arise in the variable step size implementation of high-order peer methods.

In Section 5 we present two peer methods and their properties like error constants and stability intervals. We test these peer methods on widely accepted test problems and compare them with a Runge–Kutta–Nyström method.

Finally we give some conclusions and an outlook for future work.

Section snippets

Formulation of the methods

Explicit two-step peer methods for first-order problems were considered in [4], [5], in parallel form also in [6], [7]. The second-order differential equationy″=f(t,y),t∈[t0,te],y(t0)=y0∈Rn,y′(t0)=y0′∈Rnmay be replaced by a system of first-order. Applying a standard peer method to this first-order system we obtainYmZm=B00BYm-1Zm-1+hmA00AZm-1Fm-1+hmR00RZmFm.The method uses time steps of length hm=tm+1-tm and its s stages are associated with solution values at off step points tmi=tm+hmci, i=1,…,s

Order and stability

Consistency of the peer methods (2), (3) is discussed by considering the local residuals obtained by substituting the exact solution y into the method:Δmi=y(tmi)-∑j=1sbijy(tm-1,j)-hm∑j=1saijy′(tm-1,j)-hm2∑j=1spijy″(tm-2,j)-hm2∑j=1sqijy″(tm-1,j)-hm2∑j=1i-1rijy″(tmj),Δ^mi=y′(tmi)-∑j=1sbˆijy′(tm-1,j)-hm∑j=1spˆijy″(tm-2,j)-hm∑j=1sq^ijy″(tm-1,j)-hm∑j=1srˆijy″(tmj),i=1,…,s.

Definition 3

A peer method has order of consistency p, ifmaxiΔmi=O(hmp+1)andmaxiΔ^mi=O(hmp+1).

Theorem 4

A peer method has order of consistency p, if the

Implementation

According to Remark 9 we have to ensure the boundedness of coefficient matrices multiplied by step size ratios. This is guaranteed if we allow only sufficiently small step size changes. This situation is analogous to solving stiff ODEs with BDF methods where stability constraints limit the step size ratios. A discussion of the implementation of these methods with variable step sizes where the step size changes must be sufficiently small can be found in [10]. Because the restrictions on step

Numerical tests

We will present the results for two selected peer methods, one sequential and one parallel. Other peer methods with orders between 6 and 9 and their results are presented in [9]. The two peer methods in this paper, v3-s3-p9-seq and v2-s4-p8-par, have turned out to be the best peer methods we have found for the integration of (1). v3-s3-p9-seq is a three-stage three-step ninth-order sequential peer method whereas v2-s4-p8-par is a four-stage two-step eighth-order parallel peer method (i.e. R=P=P^

Conclusions and outlook

In the numerical tests the sequential peer method requires far less function evaluations than RKN64 for nearly all tolerances and even in a sequential implementation the parallel peer method is competitive to RKN64 showing the potential of these peer methods. On the other hand the overhead of the peer implementation is much larger. In order to get realistic estimates we also considered the computing times for the Solar test (see Fig. 6). Only for crude tolerances a difference between the CPU

Acknowledgement

The authors thank the referee for his valuable comments which helped to improve the presentation of the paper significantly.

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